I'm guessing that underneath all that, you were probably overwhelmed by the fact that these people did something so incredibly nice and wonderful for you, that they thought of you in this way.
I'm sure it was also overwhelming too. and they understood - you were surrounded by people, the spotlight was on you, and it was just too much. Those are some really nice teachers you have.
I wouldn't worry about your reaction - sounds like they understand. What might be a nice gesture on your part, if you haven't already thought of it, is to give them each a thank you card saying inside how much you appreciate what they did.
Yes, it's good to let out emotions sometimes. the more you let yourself express your emotions, the more you get used to what you're feeling and how to appropriately express them. Don't be afraid of them - they are part of you, they are what makes you, you. the more you get used to them, the less they will overwhelm you. After a while, at a similar event to this birthday party, you'll find yourself laughing and smiling rather than crying - unless you are crying from happiness.
I honestly used to have a similar problem when I was younger. My mum said that between the ages of 6-8 I would just start crying in a situation that I didn't want to be in. One of those situations was swimming lessons (which I loved at the time). I decided I didn't want to swim, but I was forced to, I was fine for like 10 minutes then I just burst out crying in the pool.It wasn't even very stressful on me, from what I remember.
I'm guessing that it was the pressure that was forced onto you that caused you to just vent it all out.